Nonlinear spin-motive force driven by mixed-space quantum geometry

Spin-motive force, i.e., the electric current induced by magnetization dynamics, is theoretically studied beyond the Thouless-pump paradigm. In contrast to the linear-response regime, where the induced current is purely AC, we show that spin-motive force acquires both a DC component and a second-harmonic component at nonlinear order in magnetization dynamics. We further clarify that both contributions originate from the geometric properties of electronic bands -- quantum geometry defined in the mixed parameter space $({\boldsymbol k}, {\boldsymbol m})$ spanned by electron's momentum ${\boldsymbol k}$ and magnetization ${\boldsymbol m}$. By applying the theory to a Luttinger model, we demonstrate that our mechanism yields a finite nonlinear current even in the insulating regime, and the resulting electrical signal is measurable in a conventional current-measurement setup. Our findings offer a new operating principle of AC-to-DC conversion with magnetic materials, highlighting the pivotal role of the $({\boldsymbol k}, {\boldsymbol m})$-mixed space quantum geometry in magnetization-dynamics-induced electric currents.